## 这个作业是对大米粮食信息进行分析

Econ 490, Predictive Analytics, Final Project Instructions

The goal of this project is to calibrate a model of a poor household’s demand for dietary staples using the experimental data located in the website.

The data you are using in this project was collected by Jensen and Miller (2008, AER) using a controlled experiment that randomly assigned price subsidies on rice, the dietary staple, to extreme poor consumers in Hunan, China. With this experiment, the researchers were able to estimate the price elasticity of the demand as a function of the quantity of rice consumed prior the intervention. The following graph summarizes their results.

In the x-axis of the graph, you find the percentage of calories consumed in the form of rice prior the intervention.

For instance, 0.65 in the x-axis denotes that 65% of the total caloric-intake of the household came from rice before they were treated with the experimental subsidy. The y-axis denotes the estimate demand elasticity, which is reported as a point-estimate (the dashed blue line) and an interval estimate (the dashed red line). For example,according to the graph, the point estimate of the demand elasticity for a household with a initial caloric intake of 65% from the staple is 0.51, and its 95% confidence interval estimate is [0.10,0.93].

Your objective in this project is to match the predictions of a demand model that is given to you underneath with the facts documented by Jensen and Miller. The outcome of your project must be presented as a graph that compares the prediction with the data. The previous graph shows an example of the final output, where the solid green line denotes the values that the model predicts.

Your final report has to be between 4 and 7 pages long. In your project, you must explain how you created your output graph and the implications of your results.

The Model

The demand model you will be calibrating is the following:

� = ��% − �’ + �%�'(1 − �)�̅

�’�% − �’.

Where � is the quantity of rice consumed measured in calories, �’ is the price of purchasing 1 caloric unit of rice,�% is the price index of buying calories from all other sources that are not rice, � ∈ [0,1] is the importance of satiating hunger relative to satisfying other wants, and �̅is the amount of calories that makes the feeling of hunger become unbearable. Your objective is to find values for the independent variables such that you can match the data as best as possible.

How to Create the Final Output Graph

The final output graph has two components: the data and the prediction. To elements to reproduce the data component is available in the website. So, in this section, I will mention how to reproduce the prediction component.

You may start with these first guesses: � = 0.42, �̅= 0.88. Then, remember that the definition of elasticity is

� = %∆�%∆�’=�; − �<�<�’; − �'<�'<

This definition implies that you have to use the model to recreate two states: before the experimental treatment was delivered (�<, �'<) and after (�;, �’;). For the state before, use the price �'< = 1. For the state after, you can use any reasonable price. One reasonable price could be �’; = 0.99. Therefore, you will find that %∆�’ =−0.01.

The next part is to predict one data point. A data point is a vector with two numbers: x-value and y-value. First,focus on creating an x-value. The x-values in the graph are the percentage of calories consumed through rice before the experimental subsidy is implemented. Mathematically, this is described as �<�< + �< where �< is the quantity of calories consumed from other sources different to rice before the experiment. To find the value of �<, solve this equation that represents the budget constraint of the household:

�%�< + �< = 1

Thus, after solving for �< from the budget constraint, you will find that �<�< + �<= �%�<1 − �<(1 − �%)

This means that, in order to create an x-value, you have to choose a value for �%. Start by choosing one value arbitrarily, for example, �% = 1.2140. Then, plugin all the values in the demand function. That is,�< = ��% − �'< + �%�'<(1 − �)�̅�'<�% − �'<. = 0.42(1.2140) − 1 + 1.2140(1)(1 − 0.42)0.88 1(1.2140) − 1≅ 0.605

Thus, the x-value is�<�< + �<= �%�<1 − �<(1 − �%) = 1.214(0.605) 1 − 0.605(1 − 1.214) ≅ 0.650

Now, we have to create the y-value. To create this value, we need to calculate the elasticity. Previously, we already determined that %∆�’ = −0.01. Therefore, we need to determine the %∆�. To create the x-value, we predicted �<. This implies we are just missing to predict �;. To predict �;, we need to evaluate the demand with the new price of rice; that is, �’; = 0.99. Hence,�; = 0.42(1.2140) − 0.99 + 1.2140(0.99)(1 − 0.42)0.88 1(1.2140) − 0.99. ≅ 0.601

This implies that %∆� = �; − �<�<= 0.601 − 0.605 0.605 ≅ −0.0066

Thus, the y-axis value is � = %∆�%∆�’= −0.0066−0.01 = 0.66

That is, the predicted point is (�, �) = (0.650,0.66).

To obtain the whole predicted graph, you have to repeat this process for all the values of x. The process is almost identical, the only thing that you must change is the value of �%.